UNIVERSITÀ DEGLI STUDI DI SIENA QUADERNI DEL DIPARTIMENTO DI ECONOMIA POLITICA E STATISTICA. Stefano Vannucci. Widest Choice

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1 UNIVERSITÀ DEGLI STUDI DI SIENA QUADERNI DEL DIPARTIMENTO DI ECONOMIA POLITICA E STATISTICA Stefano Vannucci Widest Choice n. 629 Dicembre 2011

2 Abstract - A choice function is (weakly) width-maximizing if there exists a dissimilarity- i.e. an irreflexive symmetric binary relation- on the underlying object set such that the choice sets are (include, respectively) dissimilarity chains of locally maximum size. Width-maximizing and weakly width-maximizing choice functions on an arbitrary domain are characterized relying on the newly introduced notion of a revealed dissimilarity relation. Jell Classification: D01,D71 Stefano Vannucci, Dipartimento di Economia Politica e Statistica, Università di Siena vannucci@unisi.it

3 1 Introduction Choice functions may be used to model choices from subsets of jointly available objects- as opposed to alternative options- and that usage is sometimes advertised denoting them as combinatorial choice functions (see e.g. Echenique (2007), and Barberà, Bossert, Pattanaik (2004)). The analysis and assessment of biodiversity preservation policies is a prominent case in point: typically, some diversity index is to be maximized under appropriate constraints, and such a diversity index is usually meant to result from the aggregation of binary dissimilarity comparisons between the available objects, combinations of which may be chosen (see e.g. Weitzman (1998), and Nehring, Puppe (2002)). Furthermore, if dissimilarity comparisons are dichotomous (i.e. any two objects either are dissimilar or are not, with no intermediate degrees of dissimilarity) and unsupplemented by further relevant information, diversity is arguably to be assessed in terms of maximum number of mutually dissimilar objects or width (see Basili, Vannucci (2007)). Under those circumstances, diversity-maximizing combinatorial choice amounts to widest choice according to the relevant dissimilarity relation. But in fact, the range of widest choice is likely to be much larger than that. On digiting widest choice on Google one is currently o ered a quite impressive list of some 1: items: at the very least, that fact suggests that very many companies expect that their prospective customers might choose among them relying on a width-maximizing criterion. But then, how can one tell widest choice from other sorts of combinatorial choice behaviour? Is there any observable prediction following from the hypothesis of width-maximizing choice? The present note addresses the foregoing issue by providing a characterization of width-maximizing choice on a general domain, to the e ect of covering the case of nite sets of observations (see Bossert, Suzumura (2010) for a recent extensive work on maximizing choice behaviour on general domains according to an arbitrary binary relation). It will be shown that strict or irredundant width-maximizing choice rules do in fact entail some signi cant restrictions on combinatorial choice behaviour, while redundant width-maximizing choice only requires nonemptiness of the choice set at any nonempty set. 1

4 2 Model and results Let X be a universal set of items, with cardinality #X 3, and P(X) its power set. A choice function on X with domain D P(X) is a (partial) contracting operator on X i.e. a function f : D! P(X) such that f(a) A for any A 2 D. We denote by C X(D) the set of all choice functions on X with domain D. We also denote by C X the set of all choice functions on X and de ne a partial order 6 on C X by pointwise set-inclusion as follows: for any f : D! P(X) and g : D 0! P(X); f; g 2 C X, f 6 g i D D 0 and f(a) g(a) for each A 2 D. A binary relation R X X is irre exive i (x; x) =2 R for any x 2 X, and symmetric i for any x; y 2 X, (x; y) 2 R entails (y; x) 2 R, and is said to be a dissimilarity (or orthogonality) relation on X i it is both irre exive and symmetric. We also denote -for any A X- R ja = R \ (A A) and A = f(x; x) : x 2 Ag. Let D X X be 8a dissimilarity relation on X. Then, for9any A X, < B A : D jb = B B r B and = let MC(D ja ) =: #B #B 0 for any B 0 A such that : D jb 0 = B 0 B 0 ; r B 0 the set of all maximal D-chains in A. A choice function f 2 C X(D) is rationalizable by width-maximization through D i f(a) 2 MC(D ja ) for all A 2 D, and weakly rationalizable by width-maximization through D i for any A 2 D there exists B 2 MC(D ja ) such that B f(a). A choice function f 2 C X(D) is rationalizable by width-maximization (WM-rationalizable) i there exists a dissimilarity relation D X X such that f is rationalizable by width-maximization through D. Similarly, a choice function f 2 C X(D) is weakly rationalizable by width-maximization (weakly WM-rationalizable) i there exists a dissimilarity relation D X X such that f is weakly rationalizable by width-maximization through D. To put it in other, equivalent, terms WM-rationalizable (weakly WM-rationalizable) choice functions model irredundant (redundant, respectively) width-maximization. Notice that, by de nition, f 2 C X(D) is weakly WM if and only if there exists a WM f 0 2 C X(D) such that f 0 6 f. For any f 2 C X(D) its revealed dissimilarity relation D f X X may de ned by the following rule: for any x; y 2 X, xd f y i x 6= y and there exists A 2 D such that fx; yg f(a). It is immediately checked that D f is indeed irre exive and symmetric: it will play a pivotal role in the ensuing analysis. 2

5 Let us rst address the case of WM-rationalizable choice functions. The following properties are to be introduced De nition 1 (Properness (PR)) A choice function f 2 C X(D) is proper i f(a) 6=? for any A X, A 6=?. PR is a quite natural property for width-maximizing choice. Even in the worst case, when any object appears to be similar to any other objects, something -as opposed to nothing- should expectedly be chosen among the available objects (possibly a single item, if irredundance is required). De nition 2 (Revealed-dissimilarity coherence (RDC)) A choice function f 2 C X(D) is revealed-dissimilarity coherent i for all A 0 2 D and A; B X with A [ B A 0, if #A < #B and there exist fb 0 ig i2i with B 0 i 2 D for each i 2 I, such that B B [ i2i f(b 0 i) f(b 0 i) then f(a 0 ) 6= A. RDC is also a natural requirement for width-maximizing choice, at least when its revealed dissimilarity is the underlying dissimilarity relation: indeed, RDC simply dictates that the choice set of any available set of objects cannot be any subset that happens to be of smaller size of another available subset consisting of objects that are revealed to be mutually dissimilar. It may not be clear at the outset that any width-maximizing choice function should satisfy RDC. That is however one of the consequences of the following Lemma 3 Let f 2 C X(D). Then f is WM-rationalizable i it is WMrationalizable with respect to D f. Proof. Let f 2 C X(D) be WM-rationalizable through D. If D =? (notice that the empty relation is trivially irre exive and symmetric) it follows, by de nition, that 8 < MC(D ja ) =: : B A :? = B B r B and #B #B 0 for any B 0 A such that? = B 0 B 0 r B = ;

6 hence MC(? ja ) = ffxg : x 2 Ag and #f(a) = 1 for all A 2 D i.e. f is single-valued. Thus, by de nition of D f, D f =? = D hence f is trivially WM-rationalizable through D f as well. Then, suppose D 6=?, and consider any A 2 D. By hypothesis, f(a) 2 MC(D ja ) hence D jf(a) = f(a) f(a) r f(a) and #f(a) #B for any B A such that D jb = BBr B. By de nition, xd f y for each x; y 2 f(a) such that x 6= y hence D f jf(a) = f(a) f(a) r f(a). Now, suppose that there exists some B A such that #B > #f(a) and D f jb = B B r B: then, since f(a) 2 MC(D ja ), D jb B B r B hence there exist at least two distinct x ; y 2 B such that not x Dy. On the other hand, for all x; y 2 B there exists some B 0 2 D with fx; yg f(b 0 ), hence in particular there exists B 2 D such that fx ; y g f(b ) 2 MC(D jb ) whence x Dy, a contradiction since by hypothesis not x Dy. Therefore, it must be the case that #f(a) #B for all B A such that D f jb = B B r B. It follows that f(a) 2 MC(D ja ) for all A 2 D, i.e. f is WM-rationalizable through D f whenever it is WM-rationalizable at all. We are now ready to state the main characterization result of the present paper, namely Theorem 4 Let f 2 C X(D). Then f is WM-rationalizable i it satis es PR and RDC. Proof. Let f 2 C X(D) be WM-rationalizable. Then, by Lemma 3 it is WM-rationalizable ( through D f, i.e. for any A X, B A : D f jb f(a) 2 = B B r ) B and #B #B 0 for any B 0 A such that D f jb = B 0 B 0. r 0 B 0 Now, to see that f satis es PR, suppose that on the contrary f(a) =? for some nonempty A X. Thus, for any B A, D f jb = B B r B only if B =?, a contradiction since A 6=? and, for any x 2 A, D f jfxg =? by irre exivity of D f whence fxg fxg r fxg =? = D f jfxg by de nition. To check that RDC also holds, take any A 0 2 D, A; B X such that A [ B A 0, #A < #B, and there exist a family fbig 0 i2i with Bi 0 2 D for each i 2 I, and B B [ i2i f(b 0 i) f(b 0 i). Now, suppose that f(a 0 ) = A. 4

7 Since f is WM-rationalizable, it is in particular WM-rationalizable through D f, by Lemma 3. Therefore, f(a 0 ) = A 2 MC(D f ja[b ). However, by de nition of Df, B B [ f(bi) 0 f(bi) 0 entails B B n B D f ja[b, a contradiction since i2i A 2 MC(D f ja[b ) and #A < #B. Conversely, suppose that f 2 C X(D) satis es PR and RDC, and let A 2 D. Let us show that f(a) 2 MC(D f ja ) whence the thesis follows. First, notice that f(?) =? 2 MC(D f j? ) by de nition, while f(a) =? with A 6=? is impossible by PR. Now, suppose that A 6=? and f(a) =2 MC(D f ja ): then, since by de nition D f jf(a) = f(a) f(a) r f(a), it must be the case that there exists B A such that D f jb = B B r B and #B > #f(a) (hence #B > 2). Thus, by de nition of D f, for each pair x; y 2 B there exists B 0 i 2 D such that fx; yg f(b 0 i). But then, by RDC, f(a) 6= f(a), a contradiction. Remark 5 It is easily checked that PR and RDC are mutually independent properties. To see this, take X = fx; y; zg with x 6= y 6= z 6= x, D = ffx; yg ; Xg, and f : D! P(X) such that f(fx; yg) = fx; yg, f(x) = fxg. Clearly, f satis es PR. However, fx; ygfx; yg f(fx; yg)f(fx; yg) and f(fx; y; zg) = fxg entail a violation of RDC. Consider now f 0 : D! P(X) de ned as follows: f 0 (fx; yg) =?, f 0 (X) = fxg. Since [ B2D f(b) f(b) = f(x; x)g it follows that f 0 trivially satis es RDC. However, f 0 does not satisfy PR. Let us now turn to the characterization of weakly WM-rationalizable choice functions. Proposition 6 Let f 2 C X(D). Then f is weakly WM-rationalizable i it satis es PR. Proof. Let f 2 C X(D) be weakly WM-rationalizable. To begin with, notice that by de nition there exists a WM-rationalizable f 0 2 C X(D) such that 5

8 f 0 6 f. By Theorem 4, f 0 satis es PR hence by construction f also satis es PR. Conversely, let f 2 C X(D) satisfy PR. Therefore, there exists a singlevalued f 0 2 C X(D) such that f 0 6 f. Now, it is easily checked that any single-valued g 2 C X(D) is WM-rationalizable: to check this, take D =? which is trivially irre exive and symmetric, and observe that for any A X, x 2 A and B A such that #B 2:? jfxg = fxg fxg r fxg =? while? jb 6= B B r B. Thus, g is WM-rationalizable through the empty dissimilarity relation D =?. It follows that in particular f 0 is WM-rationalizable hence f 0 6 f implies that f is indeed weakly WM-rationalizable. A remarkable consequence of Proposition 6 is that if the available database only collects observations of nonempty choice sets (while empty choice sets are ignored or disallowed), then anything goes. Namely, any choice behaviour may be rationalized in terms of weak i.e. redundant width-maximization, under a not uncommon speci cation of the database. The foregoing trivializing e ect induced by mere choice of database type, however, does never hold for strict or irredundant width-maximizing choice behaviour that- by Theorem 4- is bound to display revealed-dissimilarity coherence as its unmistakable hallmark. 3 Concluding remarks Characterizations of both irredundant and redundant width-maximizing combinatorial choice behaviour on a general domain have been provided. It has been shown that while redundant width-maximizing choice does not entail any restriction on observable choice behaviour except for nonemptiness of the choice set, the hypothesis of irredundant width-maximizing choice provides much more stringent predictions. Results of this kind parallel and supplement some related work on characterizations of maximizing choice behaviour on general domains under an arbitrary binary relation (see e.g. Bossert, Suzumura (2010)), and of alternative choice rules on more specialized domains of alternative options (see e.g. Sen (1993), Baigent, Gaertner (1996), Brandt, Harrenstein (2011)). 6

9 4 References Baigent N., W. Gaertner (1996): Never choose the uniquely largest: a characterization, Economic Theory 8, Barberà S., W.Bossert, P.K. Pattanaik (2004): Ranking sets of objects, in S. Barberà, P.C. Hammond, C. Seidl (eds.): Handbook of Utility Theory, vol.ii, Extensions, Kluwer Academic Publishers, Dordrecht, pp Basili M., S. Vannucci (2007): Diversity as width, DEP Working Paper 500, University of Siena. Bossert W., K. Suzumura (2010): Consistency, choice, and rationality. Harvard University Press, Cambridge Mass. Brandt F., P. Harrenstein (2011): Set-rationalizable choice and self-stability, Journal of Economic Theory doi: /j.jet Echenique F. (2007): Counting combinatorial choice rules, Games and Economic Behavior 58, Nehring K., C. Puppe (2002): A theory of diversity, Econometrica 70, Sen A.K. (1993): Internal consistency of choice, Econometrica 61, Weitzman M. (1998): The Noah s ark problem, Econometrica 66,

Ordered Value Restriction

Ordered Value Restriction Salvador Barberà Bernardo Moreno Univ. Autònoma de Barcelona and Univ. de Málaga and centra March 1, 2006 Abstract In this paper, we restrict the pro les of preferences to be